Robust and Adaptive Model Predictive Control of Nonlinear Systems
2: United Technologies Research Centre, USA
3: LyondellBasell
Most physical systems possess parametric uncertainties or unmeasurable parameters and, since parametric uncertainty may degrade the performance of model predictive control (MPC), mechanisms to update the unknown or uncertain parameters are desirable in application. One possibility is to apply adaptive extensions of MPC in which parameter estimation and control are performed online. This book proposes such an approach, with a design methodology for adaptive robust nonlinear MPC (NMPC) systems in the presence of disturbances and parametric uncertainties. One of the key concepts pursued is the concept of setbased adaptive parameter estimation, which provides a mechanism to estimate the unknown parameters as well as an estimate of the parameter uncertainty set. The knowledge of nonconservative uncertain set estimates is exploited in the design of robust adaptive NMPC algorithms that guarantee robustness of the NMPC system to parameter uncertainty. Topics covered include: a review of nonlinear MPC; extensions for performance improvement; introduction to adaptive robust MPC; computational aspects of robust adaptive MPC; finitetime parameter estimation in adaptive control; performance improvement in adaptive control; adaptive MPC for constrained nonlinear systems; adaptive MPC with disturbance attenuation; robust adaptive economic MPC; setbased estimation in discretetime systems; and robust adaptive MPC for discretetime systems.
Inspec keywords: adaptive control; nonlinear control systems; discrete time systems; predictive control
Other keywords: adaptive model predictive control; constrained nonlinear systems; discretetime systems; robust adaptive economic MPC; disturbance attenuation
Subjects: Selfadjusting control systems; Nonlinear control systems; Optimal control; Discrete control systems; General and management topics
 Book DOI: 10.1049/PBCE083E
 Chapter DOI: 10.1049/PBCE083E
 ISBN: 9781849195522
 eISBN: 9781849195539
 Format: PDF

Front Matter
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1 Introduction
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The book provides a comprehensive introduction to NMPC and nonlinear adaptive control. In the first part of the book, a framework for the study, design, and analysis of NMPC systems is presented. The robustness of NMPC is presented in the context of this framework. The second part of the book presents an introduction to adaptive NMPC. Starting with a basic introduction to the problems associated with adaptive MPC, a robust setbased approach is developed. The third part of the book is dedicated to the practical realization of the adaptive NMPC methodology. An alternative approach to adaptive parameter estimation is first developed that yields a systematic setbased parameter estimation approach.The last part of the book presents a treatment of the discretetime generalization of the continuoustime algorithms proposed in the third part of the book.

2 Optimal control
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When faced with making a decision, it is only natural that one would aim to select the course of action which results in the “best” possible outcome. However, the ability to arrive at a decision necessarily depends upon two things: a welldefined notion of what qualities make an outcome desirable and a previous decision defining to what extent it is necessary to characterize the quality of individual candidates before making a selection (i.e., a notion of when a decision is “good enough”). Whereas the first property is required for the problem to be well defined, the latter is necessary for it to be tractable. The process of searching for the “best” outcome has been mathematically formalized in the framework of optimization. The typical approach is to define a scalarvalued cost function, that accepts a decision candidate as its argument, and returns a quantified measure of its quality. The decisionmaking process then reduces to selecting a candidate with the lowest (or highest) such measure.

3 Review of nonlinear MPC
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The ultimate objective of a model predictive controller is to provide a closedloop feedback that regulates to its target set in a fashion that is optimal with respect to the infinitetime problem, while enforcing pointwise constraints in a constructive manner.

4 A realtime nonlinear MPC technique
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The chapter provides a framework for implementing realtime predictive control calculations. The objective is to maximize the rate of inputoutput sampling frequency such that the controller approaches a continuoustime state feedback. This fast sampling rate is again achieved by simplifying the parameter search to involve only incremental improvements, so that the parameter values evolve incrementally as additional controller states in a dynamic feedback law. In contrast to any of the previous methods, however, our results do not require any assumption on the dimension of the parameterization, other than to assume that an initial set of feasible parameter values exists. The advantage of this approach is that the role of the adaptation mechanism is entirely reduced to that of performance improvement, and thus its performance may be arbitrarily suboptimal in terms of both the update increment and the parameterization basis, while still preserving stability and feasibility. As such, the most important focus and contribution of the result is the manner in which the optimization is posed at each instant, rather than the specifics of how it is solved.

5 Extensions for performance improvement
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The vast majority of MPC implementations are based upon the approximation as being PWC in time, making use of a zeroorder hold both in the input implementation as well as in the model predictions. From a theoretical perspective, there is nothing preventing the use of more general parameterizations to describe subarcs when implemented within a SD framework. In general, higherorder parameterizations involving increased number of parameters (per interval) are able to describe up over significantly longer intervals, resulting in an overall decrease in the number of parameters required to comparably describe. However, the openloop nature of the SD intervals requires that their duration be kept reasonably short, thereby eliminating most of the benefits associated with increasing the order of the parameterization. As such, it is rarely practical for SD control designs to make use of any parameterizations beyond the basic PWC selection.

6 Extensions for performance improvement
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In this book, we focus on the more typical role of adaptation as a means of coping with uncertainties in the system model. A standard implementation of MPC using a nominal model of the system dynamics can, with slight modification, exhibit nominal robustness to disturbances and modeling error. However in practical situations, the system model is only approximately known, so a guarantee of robustness which covers only “sufficiently small” errors may be unacceptable. In order to achieve a more solid robustness guarantee, it becomes necessary to account (either explicitly, or implicitly) for all possible trajectories which could be realized by the uncertain system, in order to guarantee feasible stability. The obvious numerical complexity of this task has resulted in an array of different control approaches, which lie at various locations on the spectrum between simple, conservative approximations versus complex, highperformance calculations. Ultimately, selecting an appropriate approach involves assessing, for the particular system in question, what is an acceptable balance between computational requirements and closedloop performance.

7 Computational aspects of robust adaptive MPC
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The control objective is to feasibly stabilize x to a given compact set, that is not necessarily robustly invariant with respect to the full uncertainty. It is assumed that a set and instantaneous cost are selected. The robust control design proposed depends on the knowledge of appropriate Lipschitz bounds for the xdependence of the dynamics f (x, u) and g(x, u), and for the penalty functions L(x, u) and W(x, O).

8 Finitetime parameter estimation in adaptive control
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In the conventional adaptive control algorithms, the focus is on the tracking of a given reference trajectory and in most cases parameter estimation errors are not guaranteed to converge to zero due to lack of excitation. Parameter convergence is an important issue as it enhances the overall stability and robustness properties of the closedloop adaptive systems. Moreover, there are control problems whereby the reference trajectory is not known a priori but depends on the unknown parameters of the system dynamics. For example, in adaptive extremumseeking control problems, the desired target is the operating setpoint that optimizes an uncertain cost function.

9 Performance improvement in adaptive control
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The FT identification method has two distinguishing features. First, the true parameter estimate is obtained at any time instant the excitation condition is satisfied, and second, the procedure allows for a direct and immediate removal of any perturbation signal injected into the closedloop system to aid in parameter estimation. However, the drawback of the FT identification algorithm is the requirement to check the invertibility of a matrix online and compute the inverse matrix when appropriate. To avoid these concerns and enhance the applicability of the FT method in practical situations, the procedure is hereby exploited to develop a novel adaptive compensator that (almost) recovers the performance of the FTI. The compensator guarantees exponential convergence of the parameter estimation error at a rate dictated by the closedloop system's excitation. It was shown how the adaptive compensator can be used to improve upon existing adaptive controllers. The modification proposed guarantees exponential stability of the parametric equilibrium provided the given PE condition is satisfied. Otherwise, the original system's closedloop properties are preserved.

10 Adaptive MPC for constrained nonlinear systems
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This chapter is inspired by References 47 and 49. While the focus in References 47 and 49 is on the use of adaptation to reduce the conservatism of robust MPC controller, this study addresses the problem of adaptive MPC and incorporates robust features to guarantee closedloop stability and constraint satisfaction. Simplicity is achieved herein by generating a parameter estimator for the unknown parameter vector and parameterizing the control policy in terms of these estimates rather than adapting a parameter uncertainty set directly.

11 Adaptive MPC with disturbance attenuation
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In general, modeling error consists of parametric and nonparametric uncertainties and the system dynamics can be influenced by exogenous disturbances as well. In this chapter, we extend the adaptive MPC framework presented in Chapter 10 to nonlinear systems with both constant parametric uncertainty and additive exogenous disturbances. Intuitively, an adaptive controller should lead to controller with better robustness properties than their nonadaptive counterpart since they use more information on the systems uncertainties. However, this is not generally the case. Under external disturbance input, adaptive controllers can lead to inferior transient behavior, infinite parameter drift, and burstiness in the closedloop system. To address these problems, parameter projection is used to ensure the estimate remains in a convex set and the parameter estimates are updated only when an improved estimate is obtained. The formulation provides robustness to parameter estimation error and bounded disturbances E V. While the disturbance set V remains unchanged over time, the parametric uncertainty set Θ is adapted in such a way that guarantees its contraction.

12 Robust adaptive economic MPC
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In this chapter, we propose the design of economic MPC systems based on a singlestep approach of the adaptive MPC technique proposed for a class of uncertain nonlinear systems subject to parametric uncertainties and exogenous variables. The framework considered assumes that the economic function is a known function of constrained system's states, parameterized by unknown parameters. The objective and constraint functions may explicitly depend on time, which means that our proposed method is applicable to both dynamic and steadystate economic optimization. A simulation example is used to demonstrate the effectiveness of the design technique.

13 Setbased estimation in discretetime systems
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This article presents new techniques for parameter identification for nonlinear dynamical discretetime systems. The methods presented are intended to improve the performance of adaptive control systems such as RTO schemes and adaptive extremumseeking systems. Using recent results on FT adaptive control, we develop alternative techniques that can be used to guarantee the convergence of parameter estimates to their true values in the presence of modelmismatch and exogenous variables. Three methods are presented. The first two methods rely on system excitation and a regressor matrix, in either case, the true parameters are identified when the regressor matrix is of full rank and can be inverted. The third method is based on a novel setbased adaptive estimation method proposed in Chapter 10 to simultaneously estimate the parameters and the uncertainty associated with the true value. The uncertainty set is updated periodically when sufficient information has been obtained to shrink the uncertainty set around the true parameters. Each method guarantees convergence of the parameter estimation error, provided an appropriate PE condition is met. The effectiveness of each method is demonstrated using a simulation example, displaying convergence of the parameter error estimation error.

14 Robust adaptive MPC for discretetime systems
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This article establishes a sound theoretical for the analysis of robust adaptive MPC control system subject to exogenous disturbances for a class of discretetime nonlinear control systems. No claims are made concerning the computational requirements of the proposed minmax approach to adaptive MPC technique. However, it is argued that a Lipschitzbased approach provides a conservative approximation of the minmax approach that retains all of the stability and robustness properties. The uncertainties associated with the parameters is handled using the setbased estimation approach for a class of discretetime nonlinear systems presented. It is shown how this setbased approach can be formulated in the context of nonlinear adaptive MPC approach for discretetime systems in the presence of parameter uncertainties and exogenous disturbances.

Back Matter
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